Discrete Ambiguities In Extracting Weak Phases fro

arXiv:hep-ph/9606207v13Jun1996CERN-TH/96-147EFI-96-20hep-ph/9606207June1996DISCRETEAMBIGUITIESINEXTRACTINGWEAKPHASESFROMBDECAYS1AmolS.DigheEnricoFermiInstituteandDepartmentofPhysicsUniversityofChicago,Chicago,IL60637andJonathanL.RosnerDiv.TH,CERN1211CHGeneva23,SwitzerlandandEnricoFermiInstituteandDepartmentofPhysicsUniversityofChicago,Chicago,IL606372ABSTRACTPhasesofelementsoftheCabibbo-Kobayashi-Maskawa(CKM)matrix,asobtainedusingdecaysofBmesonstoπ+π−,π±K∓,andπ+K0orπ−K0,areshowntohaveaclassofdiscreteambiguities.InmostcasesthesecanbeeliminatedusingotherinformationonCKMphases.PACScodes:11.30.Er,12.15.Hh,13.25.Hw,14.40.NdCERN-TH/96-68June19961TobesubmittedtoPhys.Rev.D.2Permanentaddress.1ApromisingsourceofinformationaboutthemechanismofCPviolationisthestudyofrateasymmetriesinthecomparisonofBandBdecaystospecificfinalstates.Theseasymmetriesofteninvolveunknownstrong-interactionphaseshifts.Amethodwasrecentlyproposed[1]tocircumventthisdifficultyusingtime-dependentB0andB0decaysandtime-integratedratesforB0→π−K+,B0→π+K−,andB+→π+K0orB−→π−K0.(Thelasttworatesarepredictedtobeequal.)WithinanassumptionofflavorSU(3)forstrongphaseshiftsandfordiagramsdominatedbytree(butnotpenguin)graphs,itwaspossibletoexhibitsixequationsinsixunknownsandthustodemonstratetheexistenceofsolutionsforallparametersofinterest.However,theMonteCarlomethodemployedinRef.[1]indicatedthepresenceofdiscreteambiguities.Usingnumericalmethodsinthepresentnote,weclarifytheseambiguities,andshowthattheymaybeeliminatedforthemostpartusingotherinformationalreadyknownaboutphasesoftheCabibbo-Kobayashi-Maskawa(CKM)matrix.TheamplitudesoftheprocessesB0→π+π−andB0→π−K+aredefinedasAππandAπK,whilethatforB+→π+K0isdefinedasA+.Theamplitudesforthecorrespondingcharge-conjugatedecayprocessesaredenotedbyAππ,AπK,A−,respectively.ItwasshowninRef.[1]thatonecanmeasuresixindependentcom-binationsofthefollowingsixparameters:thestrangeness-preservingtreeamplitudeT,thestrangeness-preservingand-violatingpenguinamplitudesPand˜P′,theweakphasesαandγ,andthestrongphaseδ.ThesecombinationsmaybeexpressedasA≡12(|Aππ|2+|Aππ|2)=T2+P2−2TPcosδcosα,(1)B≡12(|Aππ|2−|Aππ|2)=−2TPsinδsinα,(2)C≡Im(e2iβAππA∗ππ)=−T2sin2α+2TPcosδsinα,(3)D≡12(|AπK|2+|AπK|2)=(˜ruT)2+˜P′2−2˜ruT˜P′cosδcosγ,(4)E≡12(|AπK|2−|AπK|2)=2˜ruT˜P′sinδsinγ,(5)F≡|A+|2=|A−|2=˜P′2.(6)Here˜ru≡rufK/fπ,whereru≡|Vus/Vud|=0.23.ThequantitiesA−Caremeasuredintime-dependentratesforB0orB0→ππ,DandEbycomparingratesforB+→π−K+andB−→π+K−,andFviatheratefortheprocessB+→π+K0,whichispredictedtobedominatedbyasinglepenguinamplitudeandhencetohavethesamerateasB−→π−K0.Weconsidered[1]asetofrepresentativeCKMelementsparametrized[2]asshowninTableI,whereρandηaretherealandimaginarypartsofV∗ub/|VcdVcb|.Foreachofthesepoints,thephaseshiftsδ=5.7◦,36.9◦,84.3◦,95.7◦,143.1◦,and174.3◦werechosen.(Weshallnotbeconcernedherewithsinδ=0,asingularcaseinwhichtheaboveequationsnolongerprovidesufficientinformation.)MonteCarloresultsindicatedthattheequationssometimeshadmorethanonesolution.WehaveusedanexactnumericalmethodtoobtainallsolutionsofEqs.(1-6)forthepointsp1,p2,p3andthesixphasesδ.WeexpressthefiveobservablesA,B,C,D,E2TableI:Pointsinthe(ρ,η)planeandanglesoftheunitaritytriangle.Pointρηαβγ(deg.)(deg.)(deg.)p1−0.300.1520.06.6153.3p200.3570.719.390.0p30.360.27120.322.936.9intermsoffiveunknownsT,P,α,γ,δbysubstitutingthemeasuredvalueof˜P′=√Fandnotingthat˜rualsoiswell-measured.Thesolutionthenproceedsasfollows:•WeeliminateγfromEq.(4)andEq.(5)togetD=˜r2uT2+F−2˜ruT√Fcosδ1−E24˜r2uT2Fsin2δ!1/2.(7)Whenboththesidesaresquared,thisequationbecomesaquadraticinx≡sin2δwhosecoefficientsdependonlyonT.Foreachofthesolutionswhichisrealandliesbetween0.0and1.0(sincex=sin2δ),•weeliminateαfromEq.(1)andEq.(2)togetA=T2+P2−2TPcosδ1−B24T2P2sin2δ!1/2.(8)Whenboththesidesofthisequationaresquared,weobtainaquadraticiny≡P2whosecoefficientsinvolveonlyTandx,whichisaknownfunctionofT.Weproceedwiththosevaluesofythatarerealandpositive.•NowweknowalltheotherunknownsP,α,γ,δasexplicitfunctionsofasingleunknownT.WecannowcheckforthosevaluesofTwhichsatisfyEq.(3).WecanincreasetheaccuracyofoursolutionsasmuchaswewantbydecreasingthestepsizeinTandusingthezerocrossingalgorithm,whereasolutioncor-respondstothatvalueofTwhereincreasingTbyasmallamountchangesthesignof[L.H.S.-R.H.S.]inEq.(3).Asmanyas8solutionswerefoundforsomesetsofinputparameters.TheresultsaresummarizedinTablesII–IVforpointsp1−p3.WecalculateA−EfortheinputvaluesT=1,˜P′=1,P=˜P′rusinγ/sinα[assumingflavorSU(3)fortheinput],andtheinputstrongphasesshownintheTables.Theequationsaretheninvertedusingthemethoddescribedabovetoobtaintheoutputphases.Insomecasesthenumericalalgorithmgivestwocloselyrelatedoridenticalsetsofoutputphases;wehaveindicatedthesewithequalnumbers.Theseareprobablyidenticalsolutionsarrivedatthroughtwodifferentbranchesofthestep-by-stepmethoddescribedabove,withsmalldifferencesassociatedwithroundingerrors.Nonetheless,wefeelthispointcouldbenefitfromfurtherstudy.3TableII:Outputvaluesofweakandstrongphases,forgivenvaluesofinputstrongphases,indegrees,forthepointp1withαin=20.0◦andγin=153.3◦.δinαoutγoutδoutNotes5.720.01